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Mappings with bounded distortion and with finite distortion on Carnot groups

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References

  1. G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton Univ. Press, Princeton (1982) (Math. Notes,28).

    MATH  Google Scholar 

  2. P. Pansu, “Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un,” Ann. of Math. (2),129, No. 1, 1–60 (1989).

    Article  Google Scholar 

  3. L. Hörmander, “Hypoelliptic second order differential equations,” Acta Math.,119, 141–171 (1967).

    Article  Google Scholar 

  4. S. K. Vodop'yanov and A. D. Ukhlov, “Approximately differentiable transformations and change of variables on nilpotent groups,” Sibirsk. Mat. Zh.,37, No. 1, 70–89 (1996).

    Google Scholar 

  5. S. K. Vodop'yanov, “Monotone functions and quasiconformal mappings on Carnot groups,” Sibirsk. Mat. Zh.,37, No. 6, 1269–1295 (1996).

    Google Scholar 

  6. Yu. G. Reshetnyak, Space Mappings with Bounded Distortion [in Russian], Nauka, Novosibirsk (1982).

    Google Scholar 

  7. G. D. Mostow, “Quasi-conformal mappings inn-space and the rigidity of hyperbolic space forms,” Inst. Hautes Études Sci. Publ. Math., No. 34, 53–104 (1968).

    MATH  Google Scholar 

  8. A. Koranyi and H. M. Reimann, “Foundations for the theory of quasiconformal mappings on the Heisenberg group,” Adv. Math.,111, 1–87 (1995).

    Article  MATH  Google Scholar 

  9. J. Heinonen, “Calculus on Carnot groups,” in: Fall School in Analysis (Javäskylä, 1994), University of Javäskylä, Javäskylä, 1994, pp. 1–32.

    Google Scholar 

  10. S. K. Vodop'yanov, “Topological properties of mappings in Sobolev classes with summable Jacobian,” Dokl. Akad. Nauk,363, No. 4, 443–446 (1998).

    MATH  Google Scholar 

  11. S. K. Vodop'yanov, “Topological properties of mappings in Sobolev classes with summable Jacobian,” Sibirsk. Mat. Zh. (to appear).

  12. S. K. Vodop'yanov, “On differentiability of mappings in Sobolev classes on a Carnot group,” Sibirsk. Mat. Zh. (to appear).

  13. N. S. Dairbekov, “The morphism property for mappings with bounded distortion on the Heisenberg group,” Sibirsk. Mat. Zh.,40, No. 4, 811–823 (1999).

    Google Scholar 

  14. N. S. Dairbekov, “On mappings with bounded distortion on the Heisenberg group,” Sibirsk. Mat. Zh. (to appear).

  15. S. K. Vodop'yanov and V. M. Chernikov, “Sobolev spaces and hypoelliptic equations,” in: Trudy Inst. Math. Vol. 29 [in Russian], Inst. Math. (Novosibirsk), Novosibirsk, 1995, pp. 7–62. (English transl.: “Sobolev spaces and hypoelliptic equations,” Siberian Adv. Math., I:6, No. 3, 27–67 (1996); II:6, No. 4, 64–96 (1996).

    Google Scholar 

  16. J. Heinonen and I. Holopainen, “Quasiregular maps on Carnot groups,” J. Geom. Anal.,7, No. 1, 109–148 (1997).

    MATH  Google Scholar 

  17. S. K. Vodop'yanov and I. G. Markina, “Exceptional sets for solutions to subelliptic equations,” Sibirsk. Mat. Zh.,36, No. 4, 805–818 (1995).

    Google Scholar 

  18. C. J. Titus and G. S. Young, “The extension of interiority with some applications,” Trans. Amer. Math. Soc.,103, 329–340 (1962).

    Article  MATH  Google Scholar 

  19. Yu. G. Reshetnyak, “Space mappings with bounded distortion,” Sibirsk. Mat. Zh.,8, No. 3, 652–686 (1967).

    Google Scholar 

  20. J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Clarendon Press, Oxford etc. (1993).

    MATH  Google Scholar 

  21. G. B. Folland, “Subelliptic estimates and function spaces on nilpotent Lie groups,” Ark. Math.,13, 161–207 (1975).

    Article  MATH  Google Scholar 

  22. S. K. Vodop'yanov and N. A. Kudryavtseva, “Normal families of mappings on Carnot groups,” Sibirsk. Mat. Zh.,37, No. 2, 273–286 (1996).

    Google Scholar 

  23. S. Vodop'yanov, “Analysis on Carnot groups and quasiconformal mappings,” in: Abstracts: Ann Arbor Analysis Workshop (Ann Arbor, August, 1995), University of Michigan, Ann Arbor, 1995. pp. 5–6.

    Google Scholar 

  24. S. K. Vodop'yanov and I. G. Markina, “Classification of sub-Riemannian manifolds,” Sibirsk. Mat. Zh.,39, No. 6, 1271–1288 (1998).

    MATH  Google Scholar 

  25. S. K. Vodop'yanov and I. G. Markina, “Fundamentals of nonlinear potential theory for hypoelliptic equations,” in: Trudy Inst. Math. Vol. 31 [in Russian], Inst. Math. (Novosibirsk), Novosibirsk, 1996, pp. 100–160. (English transl.: “Foundations of non-linear potential theory of subelliptic equations,” Siberian Adv. Math., I:7, No. 1, 1–31 (1997); II:7, No. 2, 1–46 (1997).)

    Google Scholar 

  26. L. Capogna, “Regularity of quasi-linear equations in the Heisenberg group,” Comm. Pure Appl. Math.,50, 867–889 (1997).

    Article  MATH  Google Scholar 

  27. Yu. G. Reshetnyak, Stability Theorems in Geometry and Analysis [in Russian], Sobolev Institute Press, Novosibirsk (1996).

    MATH  Google Scholar 

  28. P. Hajłasz and J. Malý, “Approximation of nonlinear expressions involving gradient in Sobolev spaces” (to appear).

  29. S. K. Vodop'yanov, Function-Theoretic Approach to Some Problems of the Theory of Space Quasiconformal Mappings [in Russian], Avtoref. Dis. Kand. Fiz.-Mat. Nauk, Novosibirsk (1975).

    Google Scholar 

  30. S. K. Vodop'yanov and V. M. Gol'dshtenîn “Quasiconformal mappings and spaces of functions with first generalized derivatives,” Sibirsk. Mat. Zh.,17, No. 3, 515–531 (1976).

    MATH  Google Scholar 

  31. N. S. Dairbekov, “Mappings with bounded distortion on the Heisenberg group,” Sibirsk. Mat. Zh. (to appear).

  32. N. S. Dairbekov, “A limit theorem for mappings with bounded distortion on the Heisenberg group and a local homeomorphism theorem,” Sibirsk. Mat. Zh. (to appear).

  33. S. K. Vodop'yanov, “An isoperimetric inequality on Carnot groups and its applications,” Sibirsk. Mat. Zh. (to appear).

  34. S. K. Vodop'yanov and I. G. Markina, “Local estimates for mappings with bounded distortion on Carnot groups,” Sibirsk. Mat. Zh. (to appear).

  35. B. Bojarski and T. Iwaniec, “Analytical foundations of the theory of quasiconformal mappings in ℝn,” Ann. Acad. Sci. Fenn. Ser. AI Math.,8, 257–324 (1983).

    Google Scholar 

  36. Yu. G. Reshetnyak, “Sobolev-type classes of functions with values in a metric space,” Sibirsk. Mat. Zh.,38, No. 3, 657–675 (1997).

    MATH  Google Scholar 

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Authors
  1. S. K. Vodop'yanov
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Additional information

To the 70th anniversary of my teacher Yuriî Grigor'evich Reshetnyak.

The research was supported by the Russian Foundation for Basic Research (Grants 97-01-01092 and 96-15-96291) and INTAS-10170.

Novosibirsk. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 40, No. 4, pp. 764–804, July–August, 1999.

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Vodop'yanov, S.K. Mappings with bounded distortion and with finite distortion on Carnot groups. Sib Math J 40, 644–677 (1999). https://doi.org/10.1007/BF02675667

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